Independent factorial ANOVA

Two or more independent variables with two or more categories. One dependent variable.

Independent factorial ANOVA

The independent factorial ANOVA analyses the variance of multiple independent variables (Factors) with two or more categories.

Effects and interactions:

  • 1 dependent/outcome variable
  • 2 or more independent/predictor variables
    • 2 or more cat./levels

Assumptions

  • Continuous variable
  • Random sample
  • Normaly distributed
    • Shapiro-Wilk test
  • Equal variance within groups
    • Levene’s test

Formulas

Variance Sum of squares df Mean squares F-ratio
Model \(\text{SS}_{\text{model}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k_{model}-1\) \(\frac{\text{SS}_{\text{model}}}{\text{df}_{\text{model}}}\) \(\frac{\text{MS}_{\text{model}}}{\text{MS}_{\text{error}}}\)
\(\hspace{2ex}A\) \(\text{SS}_{\text{A}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k_A-1\) \(\frac{\text{SS}_{\text{A}}}{\text{df}_{\text{A}}}\) \(\frac{\text{MS}_{\text{A}}}{\text{MS}_{\text{error}}}\)
\(\hspace{2ex}B\) \(\text{SS}_{\text{B}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k_B-1\) \(\frac{\text{SS}_{\text{B}}}{\text{df}_{\text{B}}}\) \(\frac{\text{MS}_{\text{B}}}{\text{MS}_{\text{error}}}\)
\(\hspace{2ex}AB\) \(\text{SS}_{A \times B} = \text{SS}_{\text{model}} - \text{SS}_{\text{A}} - \text{SS}_{\text{B}}\) \(df_A \times df_B\) \(\frac{\text{SS}_{\text{AB}}}{\text{df}_{\text{AB}}}\) \(\frac{\text{MS}_{\text{AB}}}{\text{MS}_{\text{error}}}\)
Error \(\text{SS}_{\text{error}} = \sum{s_k^2(n_k-1)}\) \(N-k_{model}\) \(\frac{\text{SS}_{\text{error}}}{\text{df}_{\text{error}}}\)
Total \(\text{SS}_{\text{total}} = \text{SS}_{\text{model}} + \text{SS}_{\text{error}}\) \(N-1\) \(\frac{\text{SS}_{\text{total}}}{\text{df}_{\text{total}}}\)

Example

In this example we will look at the amount of accidents in a car driving simulator while subjects where given varying doses of speed and alcohol.

  • Dependent variable
    • Accidents
  • Independent variables
    • Speed
      • None
      • Small
      • Large
    • Alcohol
      • None
      • Small
      • Large

person alcohol speed accidents
1 1 1 0
2 1 2 2
3 1 3 4
4 2 1 6
5 2 2 8
6 2 3 10
7 3 1 12
8 3 2 14
9 3 3 16

Data

SSmodel

Variance Sum of squares df Mean squares F-ratio
Model \(\text{SS}_{\text{model}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k_{model}-1\) \(\frac{\text{SS}_{\text{model}}}{\text{df}_{\text{model}}}\) \(\frac{\text{MS}_{\text{model}}}{\text{MS}_{\text{error}}}\) 


##   speed alcohol accidents  n
## 1  much    much    7.5720 20
## 2  none    much    5.2970 20
## 3  some    much    6.5125 20
## 4  much    none    3.8880 20
## 5  none    none    2.1060 20
## 6  some    none    2.9445 20
## 7  much    some    5.5790 20
## 8  none    some    3.4435 20
## 9  some    some    4.7625 20
SS.model <- sum((exp.accidents - mean(data$accidents))^2); SS.model
## [1] 494.2205
m.k1 = mean(subset(data, speed == "none" & alcohol == "none", select = "accidents")$accidents)
m.k2 = mean(subset(data, speed == "none" & alcohol == "some", select = "accidents")$accidents)
m.k3 = mean(subset(data, speed == "none" & alcohol == "much", select = "accidents")$accidents)
m.k4 = mean(subset(data, speed == "some" & alcohol == "none", select = "accidents")$accidents)
m.k5 = mean(subset(data, speed == "some" & alcohol == "some", select = "accidents")$accidents)
m.k6 = mean(subset(data, speed == "some" & alcohol == "much", select = "accidents")$accidents)
m.k7 = mean(subset(data, speed == "much" & alcohol == "none", select = "accidents")$accidents)
m.k8 = mean(subset(data, speed == "much" & alcohol == "some", select = "accidents")$accidents)
m.k9 = mean(subset(data, speed == "much" & alcohol == "much", select = "accidents")$accidents)

n.k1 = n.k2 = n.k3 = n.k4 = n.k5 = n.k6 = n.k7 = n.k8 = n.k9 = 20
ss.m.k1 = n.k1 * (m.k1 - mean(accidents))^2
ss.m.k2 = n.k2 * (m.k2 - mean(accidents))^2
ss.m.k3 = n.k3 * (m.k3 - mean(accidents))^2
ss.m.k4 = n.k4 * (m.k4 - mean(accidents))^2
ss.m.k5 = n.k5 * (m.k5 - mean(accidents))^2
ss.m.k6 = n.k6 * (m.k6 - mean(accidents))^2
ss.m.k7 = n.k7 * (m.k7 - mean(accidents))^2
ss.m.k8 = n.k8 * (m.k8 - mean(accidents))^2
ss.m.k9 = n.k9 * (m.k9 - mean(accidents))^2

ss.model = sum(ss.m.k1,ss.m.k2,ss.m.k3,ss.m.k4,ss.m.k5,ss.m.k6,ss.m.k7,ss.m.k8,ss.m.k9)
ss.model
## [1] 494.2205

SSmodel visual

# Plot all data points
plot(accidents,
     xlab = 'participants')

# With the mean
lines(1:n,rep(mean(accidents),n),col='black',lwd=2)

# The black lines are the total variance, the deviation from the mean.
segments(1:n, mean(accidents), 1:n, accidents)

# The model predicts the accidents scores
points(1:n,exp.accidents, col='red')

p <- recordPlot()

# The black lines are the total variance, the deviation from the mean.
segments(1:n, exp.accidents, 1:n, mean(accidents), col = "red")

# Add legend to plot
legend("topleft",
       pch    = c("o"),
       col    = c("red"),
       legend = c("Full model") )

SSerror

Variance Sum of squares df Mean squares F-ratio
Error \(\text{SS}_{\text{error}} = \sum{s_k^2(n_k-1)}\) \(N-k\) \(\frac{\text{SS}_{\text{error}}}{\text{df}_{\text{error}}}\) 


v.k1 = var(subset(data, speed == "none" & alcohol == "none", select = "accidents")$accidents)
v.k2 = var(subset(data, speed == "none" & alcohol == "some", select = "accidents")$accidents)
v.k3 = var(subset(data, speed == "none" & alcohol == "much", select = "accidents")$accidents)
v.k4 = var(subset(data, speed == "some" & alcohol == "none", select = "accidents")$accidents)
v.k5 = var(subset(data, speed == "some" & alcohol == "some", select = "accidents")$accidents)
v.k6 = var(subset(data, speed == "some" & alcohol == "much", select = "accidents")$accidents)
v.k7 = var(subset(data, speed == "much" & alcohol == "none", select = "accidents")$accidents)
v.k8 = var(subset(data, speed == "much" & alcohol == "some", select = "accidents")$accidents)
v.k9 = var(subset(data, speed == "much" & alcohol == "much", select = "accidents")$accidents)
ss.e.k1 = v.k1 * (n.k1 - 1)
ss.e.k2 = v.k2 * (n.k2 - 1)
ss.e.k3 = v.k3 * (n.k3 - 1)
ss.e.k4 = v.k4 * (n.k4 - 1)
ss.e.k5 = v.k5 * (n.k5 - 1)
ss.e.k6 = v.k6 * (n.k6 - 1)
ss.e.k7 = v.k7 * (n.k7 - 1)
ss.e.k8 = v.k8 * (n.k8 - 1)
ss.e.k9 = v.k9 * (n.k9 - 1)

ss.error = sum(ss.e.k1,ss.e.k2,ss.e.k3,ss.e.k4,ss.e.k5,ss.e.k6,ss.e.k7,ss.e.k8,ss.e.k9)
ss.error
## [1] 66.34642

SSerror visual

p

# Now we can add the error of this model
segments(1:n, accidents, 1:n, exp.accidents, col='purple')

# Add legend to plot
legend("topleft",
       pch    = c("o", "-"),
       col    = c("red", "purple"),
       legend = c("Full model", "error") )

SSA Speed

Variance Sum of squares df Mean squares F-ratio
\(\hspace{2ex}A\) \(\text{SS}_{\text{A}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k_A-1\) \(\frac{\text{SS}_{\text{A}}}{\text{df}_{\text{A}}}\) \(\frac{\text{MS}_{\text{A}}}{\text{MS}_{\text{error}}}\) 


m.s1 = mean(subset(data, speed == "none", select = "accidents")$accidents)
m.s2 = mean(subset(data, speed == "some", select = "accidents")$accidents)
m.s3 = mean(subset(data, speed == "much", select = "accidents")$accidents)

n.s1 = n.s2 = n.s3 = 60
ss.s1 = n.s1 * (m.s1 - mean(accidents))^2
ss.s2 = n.s2 * (m.s2 - mean(accidents))^2
ss.s3 = n.s3 * (m.s3 - mean(accidents))^2

ss.speed = sum(ss.s1,ss.s2,ss.s3)
ss.speed
## [1] 128.1639

SSA Speed Visual

p
# Deviation of speed
points(1:n, data$means.speed,  col='green')
segments(1:n, mean(accidents), 1:n, data$means.speed, col='green')

# Add legend to plot
legend("topleft",
       pch    = c("o", "o"),
       col    = c("green", "red"),
       legend = c("speed", "Full model") )

SSB Alcohol

Variance Sum of squares df Mean squares F-ratio
\(\hspace{2ex}B\) \(\text{SS}_{\text{B}} = \sum{n_k(\bar{X}_k-\bar{X})^2}\) \(k_B-1\) \(\frac{\text{SS}_{\text{B}}}{\text{df}_{\text{B}}}\) \(\frac{\text{MS}_{\text{B}}}{\text{MS}_{\text{error}}}\) 


m.a1 = mean(subset(data, alcohol == "none", select = "accidents")$accidents)
m.a2 = mean(subset(data, alcohol == "some", select = "accidents")$accidents)
m.a3 = mean(subset(data, alcohol == "much", select = "accidents")$accidents)

n.a1 = n.a2 = n.a3 = 60
ss.a1 = n.a1 * (m.a1 - mean(accidents))^2 
ss.a2 = n.a2 * (m.a2 - mean(accidents))^2 
ss.a3 = n.a3 * (m.a3 - mean(accidents))^2 

ss.alcohol = sum(ss.a1,ss.a2,ss.a3)
ss.alcohol
## [1] 364.1458

SSB Alcohol Visual

p

# Deviation of alcohol
points(1:n, data$means.alcohol,col='orange',type="p")
segments(1:n, mean(accidents), 1:n, data$means.alcohol, col='orange')

# Add legend to plot
legend("topleft",
       pch    = c("o", "o"),
       col    = c("orange", "red"),
       legend = c("alcohol", "Full model") )

SSAB Alcohol x Speed

Variance Sum of squares df Mean squares F-ratio
\(\hspace{2ex}AB\) \(\text{SS}_{A \times B} = \text{SS}_{\text{model}} - \text{SS}_{\text{A}} - \text{SS}_{\text{B}}\) \(df_A \times df_B\) \(\frac{\text{SS}_{\text{AB}}}{\text{df}_{\text{AB}}}\) \(\frac{\text{MS}_{\text{AB}}}{\text{MS}_{\text{error}}}\) 


# Sums of squares for the interaction between speed and alcohol
ss.speed.alcohol <- ss.model - ss.speed - ss.alcohol
ss.speed.alcohol
## [1] 1.910727

Mean Squares

Mean squares for:

  • Speed
  • Alcohol
  • Speed \(\times\) Alcohol

\[\begin{aligned} F_{Speed} &= \frac{{MS}_{Speed}}{{MS}_{error}} \\ F_{Alcohol} &= \frac{{MS}_{Alcohol}}{{MS}_{error}} \\ F_{Alcohol \times Speed} &= \frac{{MS}_{Alcohol \times Speed}}{{MS}_{error}} \\ \end{aligned}\]

Interaction

\[F_{Alcohol \times Speed}\]

N          = length(accidents)
k.speed    = 3
k.alcohol  = 3
k.model    = 9
df.speed   = k.speed   - 1
df.alcohol = k.alcohol - 1
df.speed.alcohol = df.speed * df.alcohol

ms.speed.alcohol = ss.speed.alcohol / df.speed.alcohol

df.error = N - k.model
ms.error = ss.error / df.error

\(P\)-value

F.speed.alcohol = ms.speed.alcohol / ms.error
F.speed.alcohol
## [1] 1.231168
library(visualize)
visualize.f(F.speed.alcohol, df.speed.alcohol, df.error, section = "upper")

F.values = seq(0, 10, .01)

plot(F.values, df(F.values, df.speed.alcohol, df.error), type = "l", ylab="density", main = "H0 and HA F-distribution")

critical.value = qf(.95, df.speed.alcohol, df.error)

critical.range = seq(critical.value, 10, .01)

polygon(c(critical.range,rev(critical.range)), 
        c(critical.range*0, rev(df(critical.range, df.speed.alcohol, df.error, ncp = 5))), col = "green")

lines(F.values, df(F.values, df.speed.alcohol, df.error, ncp = 5))

polygon(c(critical.range,rev(critical.range)), 
        c(critical.range*0, rev(df(critical.range, df.speed.alcohol, df.error))), col = rgb(1,0,0,1), density = 40, angle = -45)

p.range = seq(F.speed.alcohol, 10, .01)

polygon(c(p.range,rev(p.range)), 
        c(p.range*0, rev(df(p.range, df.speed.alcohol, df.error))), col = rgb(0,0,1,.6), density = 20)

Contrast

Planned comparisons

  • Exploring differences of theoretical interest
  • Higher precision
  • Higher power

Post-Hoc

Unplanned comparisons

  • Exploring all possible differences
  • Adjust T value for inflated type 1 error

Effect size

General effect size measures

  • Amount of explained variance \(R^2\) also called eta squared \(\eta^2\).
  • Omega squared \(\omega^2\)

Effect sizes of contrasts or post-hoc comparisons

  • Cohen’s \(r\) gives the effect size for a specific comparison
    • \(r_{Contrast} = \sqrt{\frac{t^2}{t^2+{df}}}\)